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Properties of Convergent Sequences - Comparison Laws

We will now look at some more theorems regarding comparisons, this time with emphasis on the comparison between various sequences. If you haven't already, be sure to check out the following theorems and proofs regarding sequences:

Proof: We will show this by proof by contradiction. Let $(a_n)$ be a convergent sequence such that $\lim_{n \to \infty} a_n = A$ and suppose that $A < 0$. The $\forall \epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - A \mid < \epsilon$. This implies that:

(1)

\begin{align} A - \epsilon < a_n < A + \epsilon \end{align}

Now we note that since $A < 0$, then $-A > 0$, and so suppose we consider $-A = \epsilon_0 > 0$. Then for some $K \in \mathbb{N}$, it follows that if $n ≥ K$ then $\mid a_n - A \mid < \epsilon_0$ or equivalently: